Methods employing auxiliary sources are applied to electromagnetic scattering problems by 2-D noncircular scatterers. Specific cases are considered for which the singularities of the analytic continuation of the scattered field into the interior of the scatterer can be analytically determined. Then, it is shown that the amplitudes (currents) of the sources diverge and oscillate when the auxiliary curve does not enclose these singularities. Still, the electric fields computed by means of the divergent and oscillatory currents are convergent and correct. The correctness of the computed fields is validated by comparison with the exact solutions (where possible) as well as with the numerical solutions computed via commercial field simulation software.

We present a method for constructing superoscillatory functions the superoscillatory part of which approximates a given polynomial with arbitrarily small error in a fixed interval. These functions are obtained as the product of the polynomial with a sufficiently flat, bandlimited envelope function whose Fourier transform has at least continuous derivatives and an Nth derivative of bounded variation, N being the order of the polynomial. Polynomials of arbitrarily high order can be approximated if the Fourier transform of the envelope is smooth, i.e. a bump function.

Using an analytic-continuation argument, it is shown that the Hallén-type integral equation for the current distribution on an unloaded receiving antenna does not have a solution if the approximate kernel is used. Generalizations and consequences of this nonsolvability are discussed. The aforementioned argument does not apply when the exact kernel, which is logarithmically singular, is used.